Solve ax² + bx + c = 0 for x, with the discriminant and vertex.
Roots
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Discriminant
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Vertex
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Axis of symmetry
How this works
Quadratic formula: x = ( −b ± √(b² − 4ac) ) ÷ 2a
Discriminant: Δ = b² − 4ac
Vertex: x = −b ÷ 2a, y = f(that x)
Δ > 0: two distinct real roots. Δ = 0: one repeated real root (the parabola touches the x-axis once). Δ < 0: two complex roots (no real x-intercepts).
The vertex is the parabola's turning point — a minimum if a > 0, a maximum if a < 0.
a cannot be zero — with a = 0 the equation is linear (bx + c = 0), not quadratic.
What is the quadratic formula and how do I use it?
For ax² + bx + c = 0, the quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a. Read off a, b and c from your equation, compute the discriminant (b² − 4ac) under the square root, then apply the ± to get the two roots. Enter a, b and c above to skip the arithmetic.
What does the discriminant tell you?
Its sign tells you how many real roots exist before you even solve: positive means two real roots, zero means exactly one, negative means no real roots (the two roots are complex numbers).
What if the roots are complex?
The calculator shows them in the form p ± qi, where p is the real part (−b ÷ 2a) and qi is the imaginary part — this happens whenever the discriminant is negative.
What is the vertex used for?
It is the highest or lowest point on the parabola, and its x-coordinate (−b ÷ 2a) is also the axis of symmetry — the vertical line the parabola is mirrored around.